It would help to know the history of the camera. Was it dropped, sat
on, or at the beach? Was it working then suddenly stopped?
I've worked on a lot of digital cameras. To determine the problem with
the zoom, disassemble the camera down to the lens module. Try to
isolate the extension-retraction circuit and drive the motor from an
external source. Usually you'll find the lens has broken tracking tabs
on the main group barrel or retrofocus converter group or the ways or
guides have cracked. I recently repaired a Nikon 4100 that responded
nicely because the guide elements just snapped back into place. Usually
they snap and break off or are already broken off because most are
plastic. Some older ones are aluminum and can be bent back into shape
but that takes quite a lot of experience. Sometimes there's just a
piece of grit in the ways and you can revive it by complete disassembly
and cleaning. Otherwise you might be looking on eBay for a parts camera
with a good lens or mainboard.
Richard